ON THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD^)
BY GORDON M. FISHER Introduction. It is shown in this paper that the identity component of the group G(M) of all homeomorphisms of a closed manifold of dimension ^3 is (1) simple in the algebraic sense; (2) equal to the group of deformations of M (i.e., the group of homeo- morphisms of M isotopic to the identity homeomorphism) ; (3) open in G(M). The proofs for dimensions 2 and 3 are given separately; the proof for dimension 1, the circle, is not given here, but can be modeled after the proof for dimension 2. Theorem 9 of this paper (the main tool in obtaining the above results in the case ra= 3) is very similar toa recent independent result of Kister appear- ing elsewhere in this journal (see also [l]). It is possible to deduce from the above results that the space of homeo- morphisms of M is locally arcwise connected. However, since this is a quite special case of more general theorems of Hamstrom and Dyer [2 ] and Ham- strom [3], we omit the proof. It is also shown in this paper that if ra^3, then the group of deformations of the sphere 5„ is of index 2 in G(Sn). Furthermore, we find a characteriza- tion of homeomorphisms of degree 1 on a closed orientable manifold M of dimension ^3 which admits a homeomorphism of degree —1. Finally, we conclude as a corollary that if ra^3, then two homeomorphisms of 5B are isotopic if and only if they are homotopic. These two sets of results are related by the following fact. In showing the results of the first paragraph, we show that if M is a closed manifold of dimen- sion ^3, then a homeomorphism A of M is a deformation if and only if A = Ai ■ ■ ■ hk where A,-is a homeomorphism of M which is the identity out- side some internal closed «-cell £,• in M,n = dim M (a closed cell is internal if it lies in an open cell ; actually, we do not use all of the internal closed ra-cells in M.) In showing the results of the second paragraph, we show that if M is a closed orientable manifold of dimension ^ 3 which admits a homeomor- phism of degree —1, then a homeomorphism A of M is of degree 1 if and only
Received by the editors December 18, 1959 and, in revised form, February 22, 1960. 0) This is the author's doctoral dissertation, directed by Professor R. D. Anderson at Louisiana State University, Baton Rouge. The author would like to thank Professor Anderson for his help and encouragement. The author would also like to thank Professor Herman Meyer of the University of Miami (Florida) for various kinds of help through the years. 193
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if A = Ai • ■ • fa where A< is a homeomorphism of M which is the identity inside a closed «-cell in M, « = dim M (again we do not use all of the closed «-cells in M). The history of these problems is, to the best of my knowledge, as follows. In 1914, Tietze showed that the group of indicatrix-preserving homeomor- phisms of the 2-sphere S2 is of index 2 in the group G(S2) of homeomorphisms of S2 [4]. This was shown again by Kneser in 1926 [5] who also shows that G(Si) is locally arcwise connected (see [2]), and that these results hold also for G(5i). In 1928, Baer showed that two homeomorphisms of an orientable closed 2-manifold are homotopic if and only if they are isotopic [6; 7]. In 1934, Schreier and Ulam [8; 9] showed the index 2 result for the arc-com- ponent of the identity of G(Si) and G(S2), and they showed that the arc- component of the identity of G (Si) is simple (using results of Kneser [10 ] and Poincaré [ll]). In 1947, Ulam and von Neumann announced in an ab- stract [12] that the component of the identity of GOS2)is simple. In 1955, Fine and Schweigert studied G(7) and G(Si) (where 7/ is the closed arc) [13]. They found the normal subgroups of G(7), obtained a group-theoretic char- acterization of G(7), and proved certain theorems on the factoring of homeo- morphisms of 7/ and Si into involutions. In 1958, Anderson showed that the groups of all "orientation-preserving" homeomorphisms of 52 and 53 are simple [14]. Also in 1958, Hamstrom and Dyer showed that the group of all homeomorphisms of a compact 2-manifold is locally contractible [2]. They remark that this generalizes results of Fort [15] and Roberts [l6]. In 1959, Hamstrom announced that the space of homeomorphisms of a compact 3- manifold is LCn for every « [3]. Roberts observes in [l6] that Sanderson has proved that the group of all homeomorphisms of S3 is locally arcwise con- nected. 1. Preliminary remarks on manifolds. Let X denote a topological space, and A a subspace of X. The complement, closure, interior, and boundary of A in X are denoted by X —A, Cl(A), Int(.4), and Bndy(^4); or, when neces- sary, by Cl(A; X), etc. Let G(X) denote the group of all homeomorphisms of X, with identity e (or, when necessary, ex). Let 7?" denote the «-dimen- sional cartesian space, with the topology induced by the pythagorean metric d. For each a in 7?" and each r in R = Rl, set
Cn(a;r) = {x E R": d(x, a) = r}, On(a;r) = {* G R": d(x, a)
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G(Rn) such that A(C) = C„(0; 1). Otherwise the cell C is wild. Tame and wild open ra-cells and (ra — l)-spheres are defined similarly. Furthermore, tame and wild m-cells and (m — 1) -spheres in Rn can be defined for any mgra. It is easily verified that in 2c1, every 0-sphere and 1-cell is tame. Schoen- flies has shown [17; 18; 19] that in R2 every m-cell and (m — 1)-sphere is tame, m ^2. The result of Schoenflies on 1-spheres in R2 can be stated as follows (cf. also the Riemann mapping theorem) : Schoenflies extension theorem. Let S and 5' be 1-spheres in R2, let A be any homeomorphism of S onto S', let B and B' be the bounded components of R2 —S and R2 —S' (Jordan curve theorem), let C=SOB and C = S'OB', and let A be any 2-cell in R2 such that COC'Qlnt(A). Then there is a homeomor- phism h! of C onto C such that h'\ 5 = A, and h! can be extended to a homeomor- phism h* in G(R2) such that h*\R2—A is the identity. In R3, the situation is quite different. Antoine [20] and Alexander [21] have shown that there are m-cells and (m — 1)-spheres, 1 gm^3 (except for 0-spheres), which are wild in R3 (see also Fox and Artin [22]). Nevertheless, a partial substitute in R3 for Schoenflies' theorem can be obtained by combin- ing a result of Alexander [23] as proved by Graeub [24] (and Moise [25]), and results of Bing [26; 27; 28] which were originally obtained using results of Moise [29]. A manifold or n-manifold is here a connected separable metric space each point of which has a neighborhood (open set in the metric topology on M) whose closure in M is homeomorphic to Cn(0; 1). If ra = 0, connectedness is not required; a 0-sphere, consisting of two points, is a 0-manifold. More precisely, it is required that if x is in M, then for some neighborhood U of x, there is a coordinate homeomorphism k of CB(0; 1) onto C1(Z7). If C is a tame closed ra-cell in 2?", then there is an A in G(Rn) such that A(G) = CB(0; 1), and kh is a homeomorphism of C onto Cl(£7). Any such AA is also called a coordinate homeomorphism. The core of M, denoted by Core(M), is the space of all points of M which have a neighborhood homeomorphic to On(0; 1). The rim of M, denoted by Rim(M), is the space M-Core(M). Thus Core(M) is open in M, and Rim(M) is closed in M. (The core and rim of a manifold are traditionally called the interior and boundary; in this paper, the latter words are reserved for the point-set concepts with these names.) A manifold M is closed if it is compact and rimless (Rim(M) = 0). Cells and spheres in M are defined as in R". Let F0 be a subspace of M. A subspace F of M is tame with respect to F0 if there is an A in G(M) such that h(F) = £o- Otherwise, F is wild with respect to F0. We note, for later use, the following two easily verified facts: (1) if M is an ra-manifold, then Rim(M) (which is closed in M) is nowhere dense in M; (2) if M is connected, then so is Core(M).
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2. The smallest nontrivial normal subgroup G°(M). A closed «-cell F in an «-manifold is internal if there is an open «-cell U=k(On(0; 1)) in M such that, for some C„(d; r) in O»(0; 1), F = k(Cn(a; r)). Thus such an F lies in Core(M). (It appears to be true that any closed «-cell in Core(M) is internal* but this will not be needed in what follows.) Let $ denote the set of all internal closed «-cells in M. Let E°(M) denote the set of all A in G(M) such that, for some F in ï, A is the identity on M—F; this happens if and only if A is the identity on M-lnt(F). Such an A is the identity outside F, or supported on F, and F is the support of A. If A is in E°(M), say A is supported on F in ï, then A-1 is also supported on F, so A-1 is in E°(M). If /is in G(M), then/A/-1 is supported on /(£). The homeomor- phism/A of O„(0; 1) onto/(f7) is a coordinate homeomorphism for f(U), and fk(Cn(a; r))=f(F), so /(£) is in £. Hence/A/"1 is in £°(Af). Let G°(M) be the subgroup of G(M) generated by E°(M). Since E°(M) is closed under in- version and conjugation, G°(M) is a normal subgroup of G(M), and A is in G°(71i) if and only if A can be factored into a finite product, h —fa- • -fa, of homeomorphisms A,-of M, each supported on some £< in F. In this section it will be shown that G°(M) is the smallest nontrivial (?*e) normal subgroup of G(M), and that it is simple. Theorem 1. Let M be an n-manifold. If F is any internal closed n-cell in M, and F' is any closed n-cell in M, there is a homeomorphism f in G°(M) such thatf(F)EF'. Proof. As we remarked in §1, Core(M) is connected. The set F' of all interiors of cells of F is an open covering of Core(M). Hence, by a standard theorem of topology, there is a finite collection Vi, • • • , Vk of cells from F' such that F,nF<+i?í0, Fi meets Int(70 and Vk meets Core(F'). We may assume Fi = Int(£). Each C1(F.) is in F. Hence C1(F<)=fa(C,)Efa(0), where the ki are coordinate homeomorphisms, C,= C„(a,-; n), and O = O„(0; 1). It is easily seen that ^< = Int(Cl(F,)nCl(Fi+i))Fí0. Select 7?, = C„(tV, si) in krl(¿i)- There exists an £ = Cn(0; t),t License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 197 it follows that there is an open ra-cell Z7= A(O„(0; 2)) in Core(Af) such that Ur\h~1(U) = 0. We will construct in U a pairwise disjoint sequence of closed «-cells converging to a point of U and a homeomorphism of M which takes each ra-cell of the sequence onto the next ra-cell of the sequence. Let .4o=CB((3/8, 0, • • • , 0); 1/9). Then .40 is a closed solid ra-sphere with center on the £i-axis which is caught between the two (ra —l)-spheres 5„_i(0; 1/2) and 5„_i(0; 1/4); that is, A0 is contained in the interior of Cn(0; l/2)-Int(C»(0; 1/4)). Define a function n by n(x)=x on OB(0; 2) —Int(GB(0; 1)) and n(x) =d(x, 0)x on CB(0; 1) (here 2?" is regarded as a vector space over R). Set Ai+i = ri(Ai), i^O. The sequence A0, Ai, • • • of closed solid ra-spheres is pairwise disjoint and converges to the origin 0 of Rn. One verifies that ri is a homeomorphism of On(0; 2). Set £ = A(Cn(0; 1)), Bi = k(Al), and 23= UB<.Then 230,Bu ■ ■ ■ is a pair- wise disjoint sequence of closed ra-cells in Int(£), and the homeomorphism defined by r2 = AriA-1 on £ and r2 = identity on M— Int(£) is in £°(M), and satisfies Bi+i = r2(Bi), i^O. It will be convenient to set r = r2l. Then r is in E°(M) and r(Bj+i) =Bit t^O, so r takes each ra-cell in the sequence Bi, B2, - ■ • onto the ra-cell just preceding it in the sequence. Furthermore, r(Bo) misses B completely. Let g be any homeomorphism of M supported on the cell B 0. Then r~lgr{ is supported on r_i(B0) =25,-, i^O. Define ci by (p\Bi = r~igri\Bi and ci = identity on M— Int(23). In particular, t/>|23o= g|230. One verifies that License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 198 G. M. FISHER [November is different, and in fact w\ 730= g| 730. For, ~1leaves r(B0) pointwise fixed since it is supported on 73, and r-1 takes r(B0) back onto 730 (with g "left on it"). Hence, since w and g are both supported on 730and w|73o = g|730, w = g. Take any/ in E°(M), say/ is supported on the internal closed «-cell F. By Theorem 1, there is a homeomorphism / in G°(M) such that t(F)EBo- The homeomorphism tft~l of M is supported on t(F), therefore on 730. Hence, as we have just shown, tft~l = (r~lfa'yh~'ifa)(r~lhr)h'~l( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 199 pAp-1 and e. Hence D(X) is a normal subgroup of G(X). The second state- ment of the theorem now follows from Theorem 3. 3. Deformations of a 1-sphere or closed 2-manifold. It is easily verified that every 1-manifold is triangulable. It is known that every 2-manifold and 3-manifold is triangulable (Rado [32], Gawehn [33], Moise [29], Bing [26; 28]). These facts will be used throughout the rest of this paper. If X is a compact space, F is a metric space with metric d, and/ and g are (continuous) maps of X into Y, then p(f,g) = lnh{d(f(x),g(x):xQX} defines the Frêchet metric in the set of all maps of X into Y. If X= Y, then the set G(X) of homeomorphisms of X is a topological group in the topology induced by p (see, e.g., [3]). We will need the following theorem, which can be thought of as an exten- sion of the theorem of Schoenflies quoted in §1. A proof of this theorem can be found in [2, Lemma 2], or can be obtained using conformai mapping theory. The referee has informed me that Roberts has also given a proof of this theorem (mentioned by him in [16]). Theorem 5. Let dA and dB be two circles in R2 with center at the origin 0 bounding the closed discs A and B, where AQB. For every r>0, there is an s>0 smcAthat, given any homeomorphism h of A into B for which d(x, h(x)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 200 G. M. FISHER [November closed stars in T<-2)of the vertices of T (the closed star S[2) in Tf1 of a vertex Vi of T is the union of all closed simplexes of Ti2) which contain »<). Let r > 0 be the least distance between any two components of T'* — V, where T{ is the 1-skeleton of T' (the set of closed 1-simplexes in T'). We assert that any/ in G(T*) satisfying/! V = identity and p(f, e) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 201 d'(htpi(x), (¡>i(x)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 202 G. M. FISHER [November D(M)ECe(M). By Theorem 4, G°(M)ED(M). Therefore G*(M)=D(M) = C.(M)- 4. Deformations of a closed 3-manifold. The terms locally tame, locally polyhedral, and piecewise linear used in this section are defined in [26]. In place of the Schoenflies extension theorem of §1, and Theorem 5 of §3, the following three theorems will be used: Alexander's extension theorem [23] (as proved by Graeub [24, Theo- rem 1, §5]). Let S and S' be polyhedral 2-spheres in a triangulated 3-manifold M which are contained in the interior of a closed polyhedral 3-cell A in M, let B and 73' be the rimless components of A—S and A—S' (Jordan-Brouwer theorem), and let C=SV)B and C' = 5'U73'. There is a piecewise linear homeo- morphism h' of C onto C such that h'\ S = h, and A' can be extended to a homeo- morphism A* in G(M) such that A*| M —A is the identity. Note. Graeub only shows that A can be taken to be a 3-simplex. How- ever, if A is polyhedral, then lnt(A) can be taken piecewise linearly onto 7?3, a 3-simplex containing the images of C and C chosen, Graeub's version of Alexander's theorem applied, and then the whole taken back to Int(.4). Bing's extension theorem [26]. Let M be a triangulated 3-manifold, C a closed subspace of M, K a locally tame closed subspace of M such that K is locally polyhedral at each point of KC\C, and Note. Sanderson proves there is a simplicial s-isotopy which takes h(L) pointwise onto L and is the identity on M— U. If / is the end-stage of such an isotopy, fhix) =x for all x in L. We take g=f~l. Theorem 8. Let M be a closed 3-manifold. There is a number s>0 such that, if h is in GiM) and pih, e) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 203 hood, N is polyhedral in M (with respect to T). Let V be the subdivision of T obtained by taking the join of each barycenter of a 3-simplex of T with the vertices of that simplex. It follows from the definition of tubular neighbor- hood that the components of T2*— N are the interiors of a disjoint collection of closed polyhedral 2-cells in M, each lying in a face of some tetrahedron of T. Let ii>0 be the least distance between components of T2*— N, let <2>0 be the least distance between a component of T* —N and a 2-simplex of T2 not in T2, and set / = min {tu t2 ). Let A be any homeomorphism in G(M) such that p(A, e) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 204 G. M. FISHER [November actly three discs. Let tíi>0 be the least distance, for any s1 in T\, between a middle disc of an sl and its two end discs. Let tí2>0 be the distance from Bndy(TVi) to Bndy(TV2). Let 7i= min{«i, tí2}, and take any homeomorphism A in G(M) such that A is the identity outside TViand p(h, e) ~1fh.Since /-1$ is the identity out- side U, it can be factored into a finite product of homeomorphisms, each sup- ported on some one of the 3-cells V; hence /_10 is in G°(M). Since c/r"1/ is A-1 on each Dy, fa~lfh is the identity on each Dy. Consider the sets P which are obtained by running out from a vertex v of T, along each edge of T which meets this vertex, until we come to the Dy corresponding to this edge, and then taking the union D of these sets Dy together with that part of Ni —D which contains v. It follows from the definition of tubular neighbor- hood that the sets P are 3-cells. Now, since License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMSOF A MANIFOLD 205 linear homeomorphism of BndyiA7!) into M such that p(g, e) Theorem 9. Let M be a closed 3-manifold. The identity component C.(M) of the group G(M) of all homeomorphisms of M is simple, open in G(M), and equal to (1) the group G°(M) of all A in G(M) such that A= Ai • • • A*where hi is supported on an internal closed 3-cell in M, and (2) the group D(M) of all deformations of M (all A in G(M) isotopic to the identity e). Proof. Same as Theorem 7, using Theorem 8 instead of Theorem 6. 5. The group G(S„), ra^3. Let M be an ra-manifold, and let E'(M) denote the set of all A in G(M) such that A is the identity inside some closed ra-cell in M (not necessarily internal). Let G'(M) be the subgroup of G(M) gener- ated by E'(M). Just as in §2, one verifies that G'(M) is the set of all A in G(M) such that A= Ai • • • A*for some hi in G(M) such that A,-is the identity inside some closed ra-cell £,■ in M. The following lemma is a consequence of the Alexander and Bing exten- sion theorems quoted in §4. Theorem 10. Let M be a manifold, dim M ^¡3. There is an internal closed n-cell Fo in M, ra = dim M, such that, for any A in G(M), there is anfin G°(M) such that f(Fo) =A(£0) (setwise). Proof. The cases ra = 0 and ra = l are elementary. If ra = 2, the theorem is true for any internal closed 2-cell in M ; this follows from the Schoenflies extension theorem of §1, together with a chain argument similar to that in Theorem 1. Let dim M=3, and let M be triangulated by the locally finite complex T. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 206 G. M. FISHER [November Let F0 be an internal closed 3-simplex in some barycentric subdivision F(<) of T (the open 3-cell in which F0 lies need not be polyhedral). Take an A in G(M) and let s and s' be closed 3-simplexes in the barycentric subdivision r License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 207 A clearer view of the geometric meaning of P(F) and Q(F) can be obtained by applying Theorems 7 and 9, and replacing the groups G°(M) and G°(B) by the deformation groups D(M) and D(B). Thus A is in P(F) if and only if the ra-cell h(F) (ra = dim M) can be "slid back" by an/-1 pointwise onto F (i.e., f~%(x) =x for all x in F) ; and A is in Q(F) if and only if the ra-cell A(£) can be "slid back" setwise by an/-1 onto F, and/-1A|23 is a deformation of the (ra—l)-sphere bounding F. Theorem 11. Let M be a manifold, dim M^3, and let F0 be a pivot cell in M. For each F in T(F0), P(F) is a normal subgroup of G(M). For each F in T(Fo),P(F)=P(Fo). Proof. Take any F in T(F0) and any A in P(F). By definition, there is an / in G°(M) such that f\ F=h\F. Hence /-!A| F = identity. Since G°(M) is normal in G(M), A"1/-^ is in G°(M). Since hrlf~lh\ F = h~l\ F, A"1is in P(F). Take A, g in P(F). Then, as we have just shown, A-1, g_1 are in P(F). Hence there are p and q in G°(M) such that p\ F=h~1\ F, q\ £ = g-1| F. Since G°(M) is normal in G(M), A((gt7_1g_1)p_,)A_1 is in G°(M). Hence, since (hg)(q-1g-1)(p~1h-i)\F = hg\ F, hg is in P(M). Take any k in G(M). Since F is in T(Fo), there is a w in G(Af) such that F = w(F0). By Theorem 10, there are r and s in G°(M) such that r(£0) = w(F0) = F, and s(£0) = *w(£0) =k(F). Set f = sr~1. Then í is in G°(M) and t(F) = k(F). Since G°(M) is normal in G(M), ^((hth-^ft-^k is in G°(M). Since (k-xhk)(k-H)(h-lf)(t-lk)\ F = A_1A*|£ (because A-11F = identity), *_1A* is in P(F). This proves that P(F) is a normal subgroup of G(M). Take A in P(£0) and / in G°(M) such that /|F0 = A|F0. Since F is in T(Fo), there is a w in G(M) such that F = w(F0). By Theorem 10, there is a g in G°(M) such that g(F0) = w(Fo) = £. We have gAg"1!£=g/g_1| F. Therefore, since gfg~l is in G°(M), we have that gAg-1 is in P(F). Since, as we showed above, P(F) is a normal subgroup of G(M), we have that g~1(gAg~1)g = A is in P(F). This shows that P(F0)QP(F). Since the argument is symmetric in £o and F, we have P(F) =P(£0). Theorem 12. Let M be a manifold, dim M^3, orad /e/ F0 be a pivot cell in M. For any F in T(Fo), P(F)=G'(M). Hence an A in G(M) is in G'(M), so that h = hi ■ ■ ■ hk, hi the identity inside some closed n-cell £< in M (ra = dim M), if and only if for any n-cell Fin M tame with respect to Fo, there is a deformation f of M such that f(x) =h(x) for every x in F. Proof. Take h in E'(M), say A is the identity inside the closed cell £. By Theorem 1, there is an/ in G°(M) such that/(£0)E£- Then A is the identity inside f(F0). Hence there is a homeomorphism in G°(M), namely the identity e, such that e\f(Fo) =h\f(Fa). Hence A is in P(f(F0)). Now/(£0) ¡s in 7Y£0). Therefore, by Theorem 11, A is in P(F0). Hence, since P(£o) is a group by Theorem 11, the group G'(M) generated by E'(M) is contained in P(Fo). Take A in P(Fo) and / in G°(M) such that f\ £0 = A| F0, so that f~'h\ £0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 208 G. M. FISHER [November = identity. Since G1iM) is normal in GiM), G°(M)EGI(M) by Theorem 3. Hence / is in G'(M). Also f~% is in GZ(M) (even E'(M)), since f~lh\Fo »identity. Therefore h=ff~xh is in G'(M). Thus P(F0)EGI(M). By the previous two paragraphs, P(Fo) =GI(M). Therefore, by Theorem 11, P(F)=GI(M) for every F in T(F0). The following lemma will be needed later; the group Q(Fo) was defined just before Theorem 11. Theorem 13. Let M be a manifold, dim 7li^3, and let F0 be a pivot cell in M. For every F in T(F0), P(F) = Q(F0). Proof. Take Ain P(F0) and/ in G\M) such that /| F0 = h\ F0. Then f~%\ Bo = identity, where 730= Bndy F0, so A is in Ç(F0). Thus P(F0)EQ(F0). Take A in Q(F0) and / in G°(M) such that f(F0) = A(F0) and f~lh\ Bo is in G°(73o).By Theorem 4, /""'AIFo is in D(B0). Hence there is a family {Ht} EG(Bo) such that .CFo=/_1Ä1730, Hi = e\ Bo, and H from T30X7onto F0 defined by H(x, t) =Ht(x) is continuous. Since F0 is internal, there is a 77 such that Fo = A(C„(0; r,)EU=k(On(0; 1)) for some coordinate homeomorphism k (where n = dim M). Set C=Cn(0; r<), and take C' = C„(0; r') such that C'CO„(0; 1) and CCInt(C'). Fiber the (generalized) annulus A = C'-C by the closed arcs obtained by taking the intersection of A with each straight line through the origin of Rn. Also fiber A by the (« —l)-spheres 5„_i(0; s), r^s^r'. Define a homeomorphism , g\ Fo=f~lh\ F0, g\ M-lnt(k(C')) = identity. This definition is consistent, since g|T3o = A#A_1|F0 = AA~1TioAA~1|T3o= TTo|73o =f~1h\Bo, and setting T3= Bndy(A(C')), g| B = fabk'1\ B = kk^Hikk'- \ B = TFi| B = identity. We have/g| F0 = A| F0. Therefore, since fg is in G°(M), A is in F(Fo). Thus Q(F0)EP(Fo) =Q(F0). The theorem now follows from Theo- rem 11. Let M be an orientable closed «-manifold. In the sequel, when we speak of a homeomorphism A of degree 1 or —1 of M, we refer to the original defini- tion of Brouwer [36]. We will use this concept only in connection with orien- table closed manifolds of dimension ~3 and the «-sphere, all of which are triangulable ; hence the original definition in terms of simplicial approxima- tions can be used. When we say that such a manifold is orientable, we mean this in the sense of the theory of simplicial complexes. These concepts can, of course, also be introduced using homology theory, but we will find it more convenient to use the original definitions. We recall the following facts about the set 73(M) of all homeomorphisms on M of degree 1 (Tkfas in the last paragraph), to be used below. (1) B(M) is a normal subgroup of G(M). This follows from the fact that if/ and g are any License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 209 two maps of M into itself, then degree (fg) = degree (gf) = degree (/) • degree (g). (2) For any ra, 5„ admits a homeomorphism of degree — 1. Namely, regard Sn as the boundary of an (ra+ l)-simplex, and consider a simplicial homeo- morphism which interchanges exactly two vertices. Hence B(Sn) is a proper normal subgroup of G(Sn). (3) Since a homeomorphism of M is either of degree 1 or —1 (see, e.g., [37]), B(M) is of index 2 in G(M). (A subgroup 23 of a group A is of index 2 in A if there are exactly two left (right) cosets in .4/23. If B is normal, the "left (right)" can be omitted.) We note also the following facts about subgroups of index 2 to be used below. (1) 23 is of index 2 in A if and only if there is an r in A but not in B such that, for every A in A but not in B, A_1r is in 23; (2) if B is of index 2 in A, then there is no subgroup of A properly larger than B and smaller than A. Theorem 14. For any ra, G°(5„) =G'(Sn), and this group is a simple proper normal subgroup of G(Sn). If ra ^ 3, the index of G"(Sn) in G(Sn) is 2. Hence, for ra^3, G(5n) Aos exactly the one proper normal subgroup G°(Sn) =GI(Sn). Proof. By Theorem 3, G0(5„)CGi(5»), since G^Sn) is normal in G(Sn). If A is in £7(5B), say A is the identity in the closed ra-cell F, then there is a closed ra-simplex s in F, and 5B —Int(s) is an internal closed ra-cell in 5„. Since A is the identity inside F, A is the identity outside 5„ —Int(s). Hence A is in E°(Sn). Therefore G°(Sn)=GI(Sn), and this group is simple by Theorem 3. By Theorem 3, G°(Sn)QB(Sn). Since B(Sn) 9^G(Sn) (in fact (2) about B(Sn)), G°(S„) is proper (clearly, G°(Sn)9áe). To show that G°(S„) is of index 2 in G(Sn) for ra^3, we proceed by induc- tion. The assertion is true for the 0-sphere 5o. For, there are only two homeo- morphisms of 5o, the identity e, and the homeomorphism r which inter- changes the two points of 5o. Since r is not in £°(5o), £°(5o)=e; hence G°(5o)=e. Therefore G(5o)/G°(50) has exactly two cosets, {e} and {r}. Suppose the assertion is true for ra—1, where l^ra^3. Choose a pivot cell £0 in 5„ (Theorem 10), and set 230= Bndy£0. Since G0(Sn)QB(Sn) ^0(5«), there is a homeomorphism r of 5B not in G°(Sn). We will show that for any A in G(Sn) but not in G°(Sn), h~lr is in G°(Sn). By fact (1) about groups of index 2 noted above, this will complete the proof. Since r is not in the group G°(Sn), neither is r_1. By Theorem 10, there are /, g and A in G°(5B) such that f(Fo) = r~i(Fo), g(£o)=A(£0), and *(£») = h~Jr(Fo). By the first part of this theorem, together with Theorems 12 and 13, G°(Sn) = GI(Sn) = Q(Fo). Hence, since r"1 and A are not in Q(Fo),t1r~i\Bo and g_iA|Bo are not in G°(B0). Now Bo is an (ra —l)-sphere. Hence, by the induction hypothesis, G°(Bo) is of index 2 in G(Bo). Therefore (f-h-^BoXg-'hlBo) is in G°(230). Now A(£0) = A"1r(£„) =A-1r/-1r-1(£0) = h-hf-'r-'g-^HFo), since f-'r'^Fo) = £0 = g~lh(Fo). Moreover, A_1((r/-1r-1)g_1)A is in the normal subgroup G°(Sn) of G(Sn), since/ and g are. Hence, since G°(Sn) = G7(5n) = Q(F0), we have, by the definition of Q(F0), that *-1A-1r/-1r-1g-1A|230 = (A-1A-1r|Bo)(/-1r-1|230)(g-1A|23o) is in G"(Bo). Since, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 210 G. M. FISHER [November as we showed above, (Z-1?--1!730)(g-1A|730) is in G°(B0), we have A_1A~V|730 is in G°(T3o), since G°(T30) is a group. Thus A^r is in Q(F0) = G7(S„) = G°(Sn), as was to be shown. The group G°(S„) is, as we have seen, a proper normal subgroup of G(Sn). If TVis any proper normal subgroup of G(Sn), then G°(Sn) CTV by Theorem 3. We have shown that G°(5„) is of index 2 in G(SH) for «^3. Therefore, by fact (2) about groups of index 2 noted above, G°(5„) = TV. Theorem 15. If n —3, the following subgroups of G(Sn) are simple, open and closed in the topological group G(Sn), and equal to one another: (!) the group D(Sn) of deformations of Sn; (2) the group H(S„) of all homeomorphisms of Sn which are homotopic to the identity e; (3) the group B (Sn) of homeomorphisms of Sn of Brouwer degree 1 ; (4) the identity component Ce(Sn) of G(Sn) ; (5) the group of homeomorphisms G°(5„) = GI(Sn). Proof. By Theorem 14, G°(5„) = GI(Sn) is a proper normal subgroup of G(Sn). By Theorems 7 and 9, Ce(Sn) =D(Sn) =G°(Sn), and this group is sim- ple and open (and closed since Ce(Sn) is closed). Since Sn admits a homeo- morphism r of degree —1, B(Sn) is a proper normal subgroup of G(Sn). Since degree (e) = 1, r is not homotopic to e, by a theorem of Brouwer [36] (now a standard theorem of homology theory; see [38], where it is an im- mediate consequence of Axiom 5). Hence H(Sn) is a proper normal subgroup of G(Sn) (that it is a normal subgroup is verified as in Theorem 4). Thus, by Theorem 14, B(S„) =H(Sn) = G°(S„). Theorem 16. If n —3, then two homeomorphisms f and g of Sn are isotopic if and only if they are homotopic. Proof. If/ is isotopic to g, then/g-1 is homotopic to e, hencefg~l is isotopic to e by Theorem 15, hence/ is isotopic to g. 6. Preliminary results on the group Gl(M). Theorem 17. Let M be a manifold, dim M —3. The index of G'(M) in G(M) is =2. Proof. If G1(M)=G(M), then G'(TI7) is of index 1 in G(M). If Gl(M) 7iG(M), take an r not in GI(M). Then r_1 is not in G'(M). Take any A not in G'(M). Choose a pivot cell F0 in M (Theorem 10), and set 73o= Bndy F0. Now proceed exactly as in the induction step of the proof of Theorem 14 to show that A_1r is in GI(M); this is possible because Theorems 12 and 13 are for manifolds M, dim M^3, rather than for spheres S„, «^3. As in Theorem 14, this shows that G'(M) is of index 2 in G(M). Theorem 18. If M is an orientable closed manifold, dim 717^3, awd T17 admits a homeomorphism of degree —1, then G'(M) =73(717). That is, a horneo- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE GROUP OF ALL HOMEOMORPHISMS OF A MANIFOLD 211 morphism h of M is of degree 1 if and only if his a finite product, h = fa • • • fa, of homeomorphisms hi such that hi is the identity inside a closed n-cell F,- in 717, w= dim 717. Proof. It follows from the definition of degree as given by Brouwer [37]; (see also [38]) that any homeomorphism which is the identity inside an «-cell in 717,« = dim 717,is of degree 1. Hence EI(M)EB(M). Therefore, since 73(717) is a group (see §5), G7(T17)C73(TIT).By hypothesis, 73(717);¿ G(M). Since G'(M) and B(M) are both of index 2 in G(M) (see §5), and GJ(71T)CT3(71T), we have G'tilT) =73(717). Note. If 717is an orientable closed manifold, dim T17^2, then Tt7admits a homeomorphism of degree —1; this is easily seen by direct construction, using the classification of such manifolds into the 2-sphere and 2-sphere with handles. However, there are orientable closed 3-manifolds which do not admit a homeomorphism of degree —1 [39]. Bibliography 1. J. Kister, Isotopies in 3-manifolds with boundaries. I, Amer. Math. Soc. Notices vol. 6 (1959)Abstract 563-3, p. 642. 2. M.-E. Hamstrom and E. Dyer, Regular mappings and the space of homeomorphisms on a 2-manifold,Duke Math. J. vol. 25 (1958)pp. 521-531. 3. -, Regular mappings and the space of homeomorphisms on a 3-manifold, Amer. Math. Soc. Notices vol. 6 Part I (1959)Abstract 564-39,pp. 783-784. 4. H. 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